Remark

Tag 0CO0, permanent. Cite it as [Bourbaki, Tag 0CO0].

Commutative Algebra, chapter VIII, § 4. Séries de Hilbert-Samuel

The ring $A((T))$ is naturally identified with the ring of fractions $A[[T]]_T$ of the ring $A[[T]]$ defined by the multiplicative subset formed by the powers of $T$ .

For $n, p$ in $\mathbf{Z}$ , the natural number $\left[ \begin{array}{c} n \\ p \end{array} \right]$ is defined by

\left( \begin{array}{ll} \left[ \begin{array}{c} n \\ p \end{array} \right] = 0 & \text{if } p < 0 \text{ or } p > n, \\ \left[ \begin{array}{c} n \\ p \end{array} \right] = \binom{n}{p} = \frac{n(n-1)...(n-p+1)}{p!} & \text{if } 0 \leq p \leq n . \end{array} \right.

One has $\left[ \begin{array}{c} n \\ p \end{array} \right] = \left[ \begin{array}{c} n \\ n-p \end{array} \right]$ for $n, p \in \mathbf{Z}$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.